Priority Queueing for Packets with Two Characteristics

Transcription

1 1 Priority Queueing for Packets with Two Characteristics Pave Chuprikov, Sergey I. Nikoenko, Aex Davydow, Kiri Kogan Abstract Modern network eements are increasingy required to dea with heterogeneous traffic. Recent works consider processing poicies for buffers that hod packets with different processing requirement (number of processing cyces needed before a packet can be transmitted out) but uniform vaue, aiming to maximize the throughput, i.e., the number of transmitted packets. Other deveopments dea with packets of varying vaue but uniform processing requirement (each packet requires one processing cyce); the objective here is to maximize the tota transmitted vaue. In this work, we consider a more genera probem, combining packets with both nonuniform processing and nonuniform vaues in the same queue. e study the properties of various processing orders in this setting. e show that in the genera case natura processing poicies have poor performance guarantees, with inear ower bounds on their competitive ratio. Moreover, we show severa adversaria ower bounds for every priority queue and even for every onine poicy. On the positive side, in the specia case when ony two different vaues are aowed, 1 and, we present a poicy that achieves competitive ratio ( ) 1 + +, where is the maxima number of required processing cyces. e aso consider copying costs during admission. I. INTRODUCTION Modern networks require impementation of advanced economic modes that can be represented by desired objectives, network topoogy, buffering architecture, and its management poicy. The current Internet architecture is mosty buit for fairness, whie consideration of other objectives such as network utiization, throughput, profit and others is required [7], [3]. For a given network topoogy and buffering architecture, design of management poicies that optimize a desired objective is extremey important; a management poicy of a singe network eement incudes admission contro and scheduing poicies. Admission contro is one of the critica eements of management poicy. Most admission contro poicies are based on a simpe characteristic such as buffer occupancy, whereas traffic has additiona important characteristics such as processing requirements or vaue that are either not taken into account at a or a separate queue is aocated per traffic type. Incorporation of new characteristics (e.g., required processing per packet) in admission decisions and impementation of A previous version of this work has appeared at INFOCOM 015. Compared to the conference version, we have added severa nove resuts incuding genera ower bounds shown in Section, in particuar, a genera adversaria ower bound in Theorem 10, significanty extended the discussion parts of the paper, added Figure 4 and pubished software used for experiments. P. Chuprikov is with the Stekov Mathematica Institute at St. Petersburg and IMDEA Networks Institute, Madrid, Spain; e-mai: pschuprikov@gmai.com. S.I. Nikoenko is with the Stekov Mathematica Institute at St. Petersburg; e- mai: sergey@ogic.pdmi.ras.ru. A. Davydow is with the Stekov Mathematica Institute at St. Petersburg; e-mai: adavydow@gmai.com. K. Kogan is with IMDEA Networks Institute; e-mai: kiri.kogan@imdea.org. additiona objectives beyond fairness ead to new chaenges in design and impementation of traditiona network eements. In this work, we consider a singe-queue switch where a buffer of size B is shared among a types of traffic. e do not assume any specific traffic distribution but rather anayze our switching poicies against adversaria traffic using competitive anaysis [6], [35], which provides a uniform throughput guarantee for onine agorithms under a possibe traffic patterns. An onine agorithm ALG is α-competitive for some α 1 if for any arriva sequence σ the tota vaue transmitted by ALG is at east 1/α times the tota vaue transmitted in an optima soution obtained by an offine cairvoyant agorithm (denoted OPT). If an onine agorithm is not α-competitive for any constant α independent of the input, it is said to be non-competitive. Note that a ower bound on the competitive ratio can be proven with a specific hard exampe whie an upper bound represents a genera statement that shoud hod over a possibe inputs. In practice, the choices of processing order, impementation of push-out mechanisms etc. are ikey to be made at design time. From this point of view, our study of worst-case behaviour aims to provide a robust estimate on the settings that can hande a possibe oads. The purpose of this work is to study the impact of both packet vaues and required processing on weighted throughput; to the best of our knowedge, this is the first attempt to study such impact. The paper is organized as foows. In Section II, we formay introduce the mode we wi use in this work, a mode with both required processing and vaues. In Section III, we survey previous work in reated buffer processing agorithms. In Section I, we introduce severa agorithms based on priority queueing that appear promising for this setting; these agorithms differ in the way how they order packets: by required processing, by vaue, or by a ratio of these numbers (i.e., by vaue per one processing cyce). In Section I we begin with a negative resut: we show that a of these agorithms have at east inear competitive ratio in the genera case. Moreover, in Section we proceed to show a genera ower bound for any onine agorithm proven in an adversaria fashion; this is an important new resut for this mode as previousy considered specia cases (uniform vaues with heterogeneous processing and uniform processing with variabe vaues) aowed for optima onine poicies. However, in Section I we introduce an important specia case when there are ony two different possibe vaues, i.e., packets may have different required processing but their vaue is imited to 1 and. The maxima number of required processing cyces is. In our main resut, we present a poicy based on a priority queue that achieves competitive ratio ( )

2 Note that whie it may appear suspicious to compare packet vaues with required processing, in fact we are comparing ratios of the most vauabe (resp., heaviest) packet to the east vauabe (resp., ightest) packet because the minima required processing and minima vaue are aways set to 1. In Section II, we consider the β-push-out case, which takes copying cost into account by introducing additiona penaties for push-out (a detaied expanation of β-push-out is given in Section II). Section III presents simuation resuts where the proposed agorithms are evauated with synthesized traces, and Section IX concudes the paper. II. MODEL DESCRIPTION e use a mode simiar to the one introduced in [1], [0] and subsequenty used in [14] [16], [3] [5]. Consider a singe queue that is abe to hod B unit-sized packets and that handes the arriva of a sequence of packets, each of which is unitsized. A new part of the probem setting in this work is to combine two different characteristics of a packet. Namey, we assume that each arriving packet p is branded with: (1) the number of required processing cyces (required work, or weight) w(p) {1,..., }; () its processing vaue v(p) {1,..., }. These numbers are known for every arriving packet; for a motivation of why required processing may be avaiabe see [36], and vaues are usuay defined externay. Athough the vaues of and wi pay a fundamenta roe in our anaysis, our agorithms wi not need to know or in advance. Note that for = 1 the mode degenerates into a singe queue of uniform packets with nonuinform vaue, as considered in, e.g., [5], [37], whie for = 1 it becomes a singe queue of unit-vaued packets with different required processing, as considered, e.g., in [] [4]. e wi denote a packet with required processing w and vaue v by (w v), and a sequence of n packets with the same parameters w and v by n (w v). The queue performs three main tasks, namey: (1) buffer management, i.e., admission contro of newy arrived packets; () processing, i.e., deciding which of the currenty stored packets wi be processed; (3) transmission, i.e., deciding if aready processed packets shoud be transmitted and transmitting those that shoud. A packet is fuy processed if the processing unit has schedued the packet for processing for at east its required number of cyces. Even though a fuy processed packet is eigibe for transmission, in some settings it can be deiberatey deayed, e.g., if FIFO transmission order is required [4], [5]. e consider transmission order constraints ony once in Theorem 5 and assume that the packet is transmitted as soon as it is fuy processed. e assume discrete sotted time, where each time sot consists of three phases (see Fig. 1 for an iustration): (i) arriva: new packets arrive, and admission contro decides if a packet shoud be dropped or, possiby, an aready admitted packet shoud be pushed out; (ii) processing: one packet is seected for processing by the scheduing unit; (iii) transmission: at most one fuy processed packet is seected for transmission and eaves the queue. If a packet is dropped prior to being transmitted (whie it sti has a positive number of required processing cyces), it is ost. A packet may be dropped either upon arriva or due to a push-out decision whie it is stored in the buffer. A packet contributes its vaue to the objective function ony upon being successfuy transmitted; note that ony one packet may be transmitted per time sot. The goa is to devise buffer management agorithms that maximize the overa throughput, i.e., the tota vaue of a packets transmitted out of the queue. For an agorithm ALG and time sot t, we denote by IB ALG (t) the set of packets stored in ALG s buffer at time sot t after arriva but before processing (i.e., the buffer state shown in the second row of Fig. 1); if the timesot is cear from the context we write simpy IB ALG. For every time sot t and every packet p currenty stored in the queue, its number of residua processing cyces, denoted w t (p), is defined to be the number of processing cyces it requires before it can be successfuy transmitted, and its vaue, denoted v(p), is the number it contributes to the objective function upon transmission. Three fundamenta properties are often used in onine agorithms. First, a poicy is caed greedy if it aways accepts packets in the queue whenever it has free space. Greedy agorithms are usuay amenabe to efficient impementation and transmit everything if there is no congestion. Second, a poicy is caed work-conserving if it is aways processing as ong as it has packets with nonzero required processing in the buffer. Third, a poicy is caed push-out if it is aowed to drop packets that aready reside in its queue; note that it does not make sense for a push-out poicy to be non-greedy in the basic setting, but in the setting of Section II where admitted packets incur nonzero copying cost this may not be the case. In what foows, we wi assume that a push-out poicies are greedy and a poicies are work-conserving. III. RELATED ORK Rich iterature has been devoted to specia cases of our mode where one characteristic is assumed to be uniform. In particuar, admission contro poicies for the case of singequeued buffers where packets with uniform processing and varying intrinsic vaue arrive have been thoroughy studied. In the case of two vaues (1 and ) and First-In-First-Out (FIFO) processing order, the works [5], [37] present a deterministic non-push-out poicy with competitive ratio ( 1 ), i.e., bounded by a constant. For the more genera case, when packet vaues vary between 1 and, the works [5], [37] prove that the competitive ratio cannot be better than Θ(og ). In [4], this upper bound was improved to + n + O(n /B). In the push-out case with two packet vaues, the greedy poicy was shown in [17] to be at east 1.8 and at most -competitive. Later, the upper bound on the greedy poicy was improved to [19]; this work aso considers the β-push-out case and proves that the greedy poicy is at east competitive.

3 3 Fig. 1. A sampe time sot of PQ w,v, PQ v, w, and PQ v/w. Poicies with memory have been considered in [3], [8], [6], [8]. Recenty, packets with required processing but with uniform packet vaues in various settings have been considered in [9], [14] [16], [3] [5]. These works aso foow the paradigm of competitive anaysis, and their main resuts usuay constitute good processing poicies that have constant or ogarithmic upper bounds on the competitive ratio. For a buffer with one queue of packets with uniform vaue, priority queue that orders packets according to their required processing is known to be optima [14]. Our current work can be viewed as part of a arger research effort concentrated on studying competitive agorithms for management of bounded buffers. Initiated in [], [17], [9], this ine of research has received tremendous attention over the past decade. A survey by Godwasser [11] provides an exceent overview of this fied. Pruhs [33] provides a comprehensive overview of a reated fied of competitive onine scheduing for server systems; however, scheduing for server systems usuay concentrates on average response time and does not aow jobs to be dropped, whie we focus mosty on throughput and aow push-out. To contro increasing queueing deays introduced by packet buffers, the bounded-deay mode with varying intrinsic vaue was introduced by Kesseman et a. [18]. In that mode, each packet is associated with a sack vaue s, which denotes a hard deadine when a packet shoud be processed. The greedy agorithm that aways processes a packet with the eariest deadine is known to be -competitive [13], [18], and the best known competitive ratio is () 1.88, as shown by Engert and estermann [7]. A recent experimenta study [34] evauated the performance of different agorithms under a compatibe deadine mode. Note that a maxima sack vaue impicity bounds a buffer size even if the buffer is theoreticay unimited. For this reason, the bounded-deay mode appears to be more attractive for competitive anaysis than the mode, where a buffer is bounded expicity. In this work, we not ony consider an expicity bounded buffer but aso take into account required processing, which has a huge impact on the performance as both our theoretica resuts and simuation study wi show. Another very interesting cass of resuts in competitive anaysis are adversaria ower bounds that hod over a agorithms. Such bounds, when they can be proven, indicate that one cannot hope to get an optima onine agorithm, and a cairvoyant offine agorithm wi aways be abe to outperform it. One we-known exampe of such a bound is the ower 4 bound of 3 on the competitive ratio of any agorithm in the mode with mutipe queues in a shared memory buffer and uniform packets (i.e., packets with identica vaue and required processing) [1], [1]. For the case of a singe queue, previous works have considered two cases: variabe vaue with uniform processing and variabe processing with uniform vaues. In both cases, a singe priority queue that orders packets with respect to the variabe characteristic (argest vaue and smaest required processing first, respectivey) is optima, so there can be no nontrivia genera ower bound regardess of transmission order. In the FIFO mode, for the case of variabe vaues and uniform processing there has been a ine of adversaria ower bounds cuminating in the ower bound of that appies to a agorithms [1], with a stronger bound of for the specia case when B = if a possibe vaues are admissibe [5], [37]. In the twovaued case, tight bounds are known: an adversaria ower bound of r = 1 ( 13 1) for any B and r = 1 ( ) 1.8 for B and an onine agorithm that achieves competitive ratio r for arbitrary B and r for B [8]. In the case of variabe processing with uniform vaues, no genera ower bounds for FIFO order are known apart from a simpe ower bound of 1 ( + 1) for greedy non-push-out poicies [4]. I. ALGORITHMS AND LOER BOUNDS The utimate goa of this entire ine of research is to choose the right buffer management poicy in every probem setting, i.e., for each possibe switch configuration and every objective. For instance, previous works have studied in detai the interreations between poicies with and without push-out, the capabiity to drop previousy accepted packets from the buffer [11], [31]; whie non-push-out poicies are simper to impement in practice, they often turn out to be non-competitive in terms of weighted throughput with ower bounds on the competitive ratio inear in probem parameters such as buffer size B, maxima possibe vaue, or maxima required processing. Note that even this is not quite the whoe story: athough non-push-out poicies are usuay ceary inferior with respect to (weighted) throughput, they can sti

4 4 come out ahead for other objectives, e.g., minimizing the tota power consumption (push-out might be a costy procedure). Sti, in this work we concentrate on the weighted throughput objective, and consider a setting where worst-case ower bounds, i.e., hard exampes, are reativey easy to come by: we have two characteristics to pay with for every packet, vaue and required processing, instead of just a singe one as in majority of previous works [11], [31]. For this reason, we concentrate on studying the best cass of natura agorithms avaiabe for the singe queue setting, and previous research indicates that priority queues with push-out are the best toos that often ead to good resuts. In particuar, Kesassy et a. showed that a singe priority queue with push-out is optima for packets with varying required processing and unit vaue [14]. Note, however, that though in previous work a singe priority queue was usuay the best agorithm, sometimes simpy optima, and the goa often was to try and achieve comparabe throughput under additiona constraints such as FIFO transmission order or mutipe separate queues; in the setting with two different characteristics it is not even cear what a priority queue sorts its packets on: if one packet has ess vaue but aso ess required processing than another, which one shoud we prefer? To capture different possibe orderings in a priority queue, we introduce the foowing definition. Definition 4.1: Let f be a function of packets, f(w, v) R, with the intuition that better packets have arger vaues of f. Then the PQ f processing poicy is defined as foows: PQ f is greedy, i.e., it accepts incoming packets as ong as there is space in its buffer; PQ f is work-conserving, i.e., it processes a packet as ong as its buffer is not empty; PQ f orders and processes packets in its queue in the order of decreasing vaues of f; PQ f pushes out a packet p and adds a new packet p to the queue at time sot t if the buffer is fu, p is currenty the worst packet in the buffer and p is better than p: f(p) = min q IB PQ f f(q), and f(p ) > f(p). Here IB PQ f is IB PQ f (t) for the current time sot t. In other words, PQ f orders and processes packets according to the function f. Note that this definition, again, restricts the space of possibe agorithms. In theory, we coud separate admission and processing order, accepting and pushing out packets with respect to one ordering but processing and transmitting them with respect to a different ordering. In [4], in the setting with one characteristic and FIFO transmission order constraint, a simiar idea decoupe transmission order from processing order has ed to significant improvements in competitive ratios, incuding a constant upper bound on the competitiveness of a poicy which admitted and processed its packets as a priority queue but transmitted them in FIFO order. However, throughout this paper we simpify our considerations and assume that admission and processing orders are the same. In particuar, we consider three specific priority queues (here w denotes the current residua work and v denotes the packet s vaue): (1) PQ w,v = PQ w+v/( +1) orders packets in the increasing order of their required processing, breaking ties by vaue; () PQ v, w = PQ v w/( +1) orders packets in the decreasing order of their vaue, breaking ties by required processing; (3) PQ v/w orders packets in the decreasing order of their vaue-to-work ratio, i.e., it prioritizes packets that yied the best vaue per one time sot of processing. 1 Fig. 1 shows a sampe time sot of these priority queues; in this case B = 3, a poicies start with (5 ), (4 3), and (1 1) in their queues, and a (6 3) packet arrives. PQ w,v rejects the (6 3) since it has the argest processing requirement, PQ v,w pushes out (1 1) since it has the smaest vaue, and PQ v/w pushes out (5 ) since it has the worst v/w ratio of /5 compared to 3/4, 1, and 3/6 of the other three packets. One of the goas of this work is to expore which order performs best. Our main resut in this part is that, in genera, priority queues fai to provide constant or even ogarithmic competitiveness in the setting with two packet characteristics, as they do in cases, where there is ony a singe characteristic. For a three specific PQ poicies shown above, we prove inear (in and/or ) ower bounds on their competitive ratios against an optima agorithm. This is an interesting and somewhat discouraging resut since priority queues have proven to be efficient when each characteristic is considered separatey, often with constant upper bounds on the competitive ratio or even shown to be optima poicies. Note that whie it may seem intuitive that PQ v/w shoud be best at east among these three, we wi see ower bounds for a of them in this section, and ater an even more counterintuitive resut in a specia case with two vaues. For a ower bound, it suffices to present a hard sequence of packets on which the optima agorithm outperforms the one in question; so in the theorems beow we simpy describe this sequence. e aso show matching upper bounds when appicabe. Theorem 1: For a buffer of size B, maxima packet required processing, and maxima packet vaue, PQ w,v is at east -competitive and at most -competitive. Proof: First, there arrive B (1 1) packets, i.e., B packets with required processing 1 and vaue 1; PQ w,v accepts them whie OPT does not. Then there arrive B ( ) packets accepted by OPT; PQ w,v skips them since they have arger required processing than aready admitted. No more packets arrive, so in B steps PQ w,v processes packets with tota vaue B; OPT, with tota vaue B. The same sequence is repeated to get the asymptotic bound. The upper bound foows since PQ is optima for uniform vaues and variabe required processing; this means that PQ w,v processes as many packets as OPT, so it cannot ose by a factor of more than. 1 Note that here we aso have two possibiities for breaking ties, PQ v/w, w = PQ v/w w/( +1) and PQ v/w,v = PQ v/w+v/( +1), but in this case the tie-breakers wi be irreevant for a our statements, so we unite them under the same notation.

5 5 Theorem : For a buffer of size B, maxima packet required processing (, and maxima ) packet vaue, PQ v, w is at east ( 1) o(1) -competitive. Proof: In the first burst, there arrive B ( ) which PQ v, w accepts but OPT does not. Then, over the next time sots there arrives a (1 1) packet on every time sot. OPT accepts, processes, and transmits this packet immediatey, whie PQ v, w drops it since it has worse vaue than the ones in its queue. On time sot +1, when PQ v, w has processed one ( ) packet, another ( ) arrives, to be accepted by PQ v, w, and this brings us back to the same state as on the first time sot. Thus, over this sequence PQ v, w has processed packets with tota vaue, OPT has processed packets with tota vaue ( 1), and the sequence can be repeated. After we repeat the sequence C times, we finish by fushing both buffers with B (1 ). Thus, both agorithms wi end with B more processed vaue, and the competitive ratio over this sequence is ( 1) C + B. (B + C) It remains to et C. So is PQ v/w that combines characteristics and aims for the best vaue per timesot any better in the worst case? Unfortunatey, no. Theorem 3: For a buffer of size B, maxima packet required processing, and maxima packet vaue, PQ v/w is at east min{, }-competitive. Proof: e denote m = min{, }. In the first burst, there arrive B (1 1) packets which PQ v/w accepts but OPT does not. Then on the same time sot there arrive B (m m) packets which OPT accepts but PQ v/w does not since they have work-to-vaue ratio worse than 1. Thus, in Bm steps OPT transmits tota vaue mb, whie PQ v/w ony transmits tota vaue B, and this sequence can be repeated.. GENERAL LOER BOUNDS Theorems 1 3 have estabished that PQ w,v, PQ v, w, and PQ v/w are non-competitive. But perhaps we have just faied to be inventive enough in designing these priority queues? Maybe we can devise a better priority queue, or simpy a better onine agorithm that wi achieve constant competitiveness or even be optima? In this section, we dash these hopes by proving genera ower bounds on a onine agorithms. They are proven in an adversaria way: we construct a sequence of inputs where further inputs depend on the choices an onine agorithm makes, so in the end we find a bad input for every possibe choice. Note that some of the bounds beow are nontrivia ony in the extreme cases of, > B and even, B, but it sti shows that we cannot hope for constant upper bounds uness we expicity assume B >, and somehow use it in the proof. On the positive side, note that most of these ower bounds ony need two kinds of packets, so they aso work in restricted settings where vaue and/or work can ony take some of the vaues in their respective intervas. Later we wi see that an important specia case is the twovaued case, when required processing can be an arbitrary integer 1 w but there are ony two possibe vaues, 1 and ; we wi prove an upper bound for this case in Section I. Therefore, we note specia cases of ower bounds for this case as we. Note that a ower bounds triviay extend from the two-vaued case to the genera case but not vice versa. e begin with a very simpe case to iustrate basic ideas. It turns out that even reducing the buffer to a singe sot does not et us construct a competitive onine agorithm. The basic idea for the foowing and most other genera ower bounds is to have ight and heavy packets such that it is x times better to process ight packets per time sot, but a heavy packet is x times better than a singe ight packet, so if the agorithm pushes out the heavy packet, we can stop the arrivas and win x times the vaue again. This is where the that appears here and in many further bounds comes from: since a heavy packet is x times worse per time sot but x times better overa, it must have x times the processing of a ight packet. Theorem 4 (B = 1): For B = 1 and >, any onine agorithm ALG is at east -competitive. Further, in the two-vaued case or if any onine agorithm ALG is at east min{, / }-competitive for B = 1. Proof: On the first step, two packets arrive, ( ) and ( 1 ). If ALG accepts ( 1 ), no other packets arrive, OPT accepts ( ) and wins by a factor of in ( vaue. If ) ALG accepts ( ), the same pair of packets, 1 and ( ), continue to arrive every tick, OPT processes ight packets and earns vaue whie ALG earns. If ALG decides to switch to a ighter packet in the process, arrivas stop immediatey, OPT accepts the current ( ), and the resut is even worse for ALG. For the case when the vaue of / ( is either ) unavaiabe or is ess than or equa to one, repace 1 with (1 1) and observe that in the first case we get a ratio of and in the second, /. In traditiona networking, most buffers impement FIFO processing order because of its simpicity and desired properties. The foowing theorem demonstrates that under the FIFO constraint on processing and transmission order, ower bounds may significanty deteriorate and become non-competitive. Note that the admission order is not constrained in the theorem, ALG is free to push out any packet. Theorem 5 (FIFO order): For arbitrary B and >, any onine agorithm ALG that ( preserves FIFO ) processing and transmission order is at east B B -competitive. In the two-vaued case or if, any onine agorithm ALG with FIFO processing and transmission order is at east +B 1 +B 1 -competitive. ( Proof: ) The construction is very simiar: ( ) and B 1 arrive on the first step. Then ALG either: (a) drops ( ), in which case arrivas stop, and OPT earns tota vaue + (B 1) whie ALG earns B, for the tota ratio of +(B 1)/ B/ = B B ; or (b) ( keeps processing ( ) whie we feed OPT with more 1 )s; then arrivas stop, ALG finishes the other B 1 of his packets and earns tota vaue + (B 1)

6 6 whie OPT has earned ( +B 1) in vaue; the ratio in this case is worse, ( + B 1)/ + (B 1)/ = + B 1 + B 1 > For the second part, repace ( ) 1 B + 1. with (1 1) and observe that in case (a) we get a competitive ratio of +B 1 B, and in case (b) the competitive ratio is +B 1 +B 1, which is smaer. Next, we turn to priority queues. e have aready mentioned that priority queues arise naturay as candidates for best possibe poicies, but, again, by restricting our consideration to the cass of priority queues (i.e., agorithms that have deterministic inear orderings on the packets) we get a arger genera ower bound, regardess of what this order specificay is. e woud ike to emphasize that not a agorithms can be represented as PQ f. For exampe, some agorithms may base their decisions on buffer occupancy or statistica data coected over previous timesots. In particuar, Theorem 4 is not a specia case of the foowing theorem. Theorem 6 (arbitrary PQ): For > and any priority function f such that f(w, v) R, the agorithm PQ f is at east -competitive. In the two-vaued case or if, agorithm PQ f is at east min{, / }-competitive. Proof: Again, ( we ony need two kinds of packets, ( ) and 1 / ). There are two cases: (1) if PQ f prefers ( ), i.e. f(, ) > f(1, / ), we keep feeding the agorithms with both kinds of packets; PQ f chooses ( and processes ( )s whie OPT is processing 1 / ) s, getting times more vaue per time sot; ( () if, on the other hand, PQ f prefers 1 / ) to ( ( ), then B 1 / ) and B ( ) arrive on the first burst and then arrivas stop; ALG fis its buffer with ight packets, OPT takes the heavy ones, and after B time sots OPT has again transmitted times more vaue. ( ) For the second part, again, repace 1 with (1 1). Finay, we consider genera ower bounds for a deterministic onine agorithms. e begin with the two-vaued case. The idea of the foowing ower bound is to send in penty of both ight packets (1 1) and heavy packets ( ). If ALG accepts few heavy packets, OPT can accept a of them, hat arrivas, and win a ot with the sequence. If ALG accepts a ot of heavy packets, OPT fuy processes a ight packets, winning over ALG that has to begin processing heavy ones, and then the buffers are fushed out with the best possibe packets (1 ). Theorem 7: For a buffer of size B, maxima packet required processing, and avaiabe packet vaues 1 and ( > 1, every onine deterministic agorithm ALG is at east O ( )) 1 -competitive. Proof: On the first step there arrive B (1 1) and B ( ). Suppose that ALG has accepted n of ( ) packets. Again, there are two cases. n < + 1B : in this case, OPT chooses to accept B ( ), and no new packets arrive. After B time sots, OPT has processed packets with tota vaue B, and ALG has processed at most (B n) + n, yieding competitive ratio 1 B B n + n = 1 + ( 1) n, B which is at east + 1 for n B < n 1 + 1B : in this case, OPT accepts B (1 1). After B time sots, OPT has transmitted tota vaue B, whie ALG has processed at most (B n) (1 1) and B/ ( ). Now B (1 ) arrive and then no more packets; after a buffers have been emptied OPT gets B more vaue and ALG at most B, which yieds the ratio ( ) B + B (B n) + B/ + B = n/b O ( ) O for n B These same ideas can ead to a genera ower bound on a deterministic onine agorithms. e first show the proof for B = 1 here and then show the proofs for B = as a characteristic specia case and then for the genera case of arbitrary B. These proofs essentiay rey on the avaiabiity of a packet whose vaue is a specific fraction of, and thus they are not appicabe in the two-vaued case. The proof for the genera case is technicay quite invoved so we consider in detai the case of B = and then show a proof sketch for arbitrary B. The basic idea of ight and heavy packets remains the same, but for buffer size B we wi need B + 1 eves of different packets to get a recursive construction and an inductive proof of the bound. This eads to 4 in the case of B = and B in the genera case. Theorem 8 (arbitrary onine ALG, B = 1): For B = 1, the competitive ratio of any deterministic onine agorithm ALG is at east min{, }. Proof: There are ony two kinds of packets invoved in the ower bound: heavy packets ( ) and medium packets ( ), where is a parameter to be defined ater. On the first burst, both packets arrive, 1 ( ) and 1 ( ) (, and then on every time sot 1 ) arrives unti ALG accepts it. Denote by t the time when ALG accepts a medium packet instead of ( ). There are two cases. 1. t =, i.e., ALG processes ( ) to competion. In this case, we repeat the sequence by sending another ( ) after time sots. OPT wi process medium packets a the time, getting tota vaue of per time sots whie ALG obtains vaue per time sots.. At some t <, ALG accepts a medium packet, pushing out the heavy one. In this case, medium packets immediatey stop, and OPT processes ony the heavy packet with vaue whie ALG processes one medium packet with tota vaue. Then both buffers become empty, and the sequence can be repeated.

7 7 e now take = min{, } to get the bound. Theorem 9 (arbitrary onine ALG, B = ): For B =, the competitive ratio of any deterministic onine agorithm ALG is at east 1 min{ 4, }. Proof: The basic idea is to preserve the foowing invariant: on every step except a sma fraction OPT wi obtain at east times more vaue than ALG. There are three kinds of ( packets in the sequence: heavy ( ), medium ) (, and ight ) 4, where is a parameter to be defined ater. On the first burst, there arrive a heavy and a medium packet, 1 ( ) and 1 ( ), foowed by two ight packets, ( ) 4. Then, on every time step two more ight packets arrive, ( ) 4. There are severa cases. Note that in what foows, OPT aways keeps one heavy packet in its buffer, and heavy packets never arrive during the period we are counting the packets in. A new heavy packet ony arrives when the sequence is repeated from the start. 1. ALG does not push out either heavy or medium packet in favor of ight ones. Then OPT keeps one heavy packet in the buffer whie it processes ight packets, getting vaue per processing cyce whie ALG is getting at most (from the medium packet). As soon as ALG finishes the medium or heavy packet, another packet of the same kind arrives, and the sequence is repeated. Note that this ratio of in favor of OPT occurs every time OPT is abe to process a ight packet whie ALG is processing a different one.. At some timesot t, ALG pushes out the heavy packet for a ight one. In this case, OPT accepts the ast arriving medium packet (note that there may have been severa medium packets arriving due to ALG processing them to competion), and there are no more arrivas after timesot t. OPT finishes the heavy packet residing in its buffer and the ast arriving medium packet whie ALG can process at most a medium and a ight one. As a resut, before time t OPT had times more vaue per timesot by case 1, and after time t ALG has earned at most ( ) tota vaue whie OPT has earned (1 + 1 ), for the tota ratio of over the entire sequence. 3. At some timesot t, ALG pushes out the medium packet for a ight one. This is a sighty more compicated case, with two subcases depending on the time t when the pushed out medium packet had arrived: (i) if t t < (the pushed out medium packet arrived ess than timesots ago), OPT accepts it at time t, and a arrivas stop unti time t +, i.e., unti OPT finishes processing it; over these timesots: OPT is getting vaue per time sot every time, for a tota vaue ; ALG can get tota vaue once by processing this ight packet, but on a other timesots it coud not get more than vaue per time sot (assuming it was processing the heavy packet); hence, over this period of time ALG gets no more than + tota vaue; hence, the competitive ratio over these timesots is at east ; (ii) if t t (the pushed out medium packet arrived at east timesots ago), OPT is processing ight packets a this time, and as soon as ALG has finished the ight packet, another medium packet arrives, reverting to the origina situation; in this case: OPT has processed (t t ) 4 ight packets pus the one fina ight packet, obtaining tota vaue at east (t t ) + ; ALG has obtained tota vaue at most (t t ) over the past t t timesots (if ALG was processing the medium packet it is now worth nothing since it has been pushed out, so vaue can ony come from the heavy packet) pus the fina ight packet, for a tota of at most (t t ) + ; since t t, the competitive ratio is at east (t t ) + (t t ) + + = + 1. As a resut, during the entire sequence the competitive ratio is never smaer than, and the constraint on is that min{ 4, }. Theorem 10 (genera case): For arbitrary B, the competitive ratio ( of any deterministic onine agorithm ALG is at east 1 min{ B, B ) } 1. Proof of Theorem 10: e proceed by induction with a construction simiar to the proof of Theorem 9. The induction base for B = 1 and B = has aready been considered in Theorems 8 and 9. For the induction step, consider k + 1 types of packets that differ by a factor of v from each other in vaue and by a factor of v in required processing: the first packet has vaue 1 and work 1, the second vaue v and work v, and so on unti vaue v k and work v k ; for this to work we have to have v min{ B, B }. In the arrivas, we wi preserve the invariant that ALG s buffer can ony have two packets of identica vaue if they are the cheapest; i.e., we wi not give repeated packets to the agorithm but send in a new packet of a certain size ony after the previous one has finished processing or has been dropped; the cheapest and ightest packets are coming in with a steady stream, and in most cases OPT keeps processing them. If ALG at some point pushes out the heaviest packet, arrivas stop immediatey, and OPT finishes the heaviest packet; in this case, ALG oses by a factor of at east 1+v+v + v k 1. If v k ALG is working on the heaviest packet, OPT is working on the ightest packet and has unit gain over ALG by a factor of at east v k 1. If, otherwise, ALG keeps the heaviest packet in the buffer and is working on some other packet, arrivas and OPT are operating as in the induction hypothesis for B 1, with one sot in the buffer reserved for the heaviest packet. There is ony one exception to this operation: if ALG has pushed out the second heaviest packet (which is heaviest in the B 1 case), arrivas do not stop competey but cease temporariy for v (k 1), i.e., for the time it takes to process the second

8 8 heaviest packet. Afterwards, we again send in one packet of each type except the heaviest. The technicaity here is that by sending in these packets to reset the buffer, we are vioating the decared invariant. To fix this, we count a packets except the heaviest one as processed by the agorithm and assume that it has gotten fu vaue for them (afterwards, if ALG is working on one of the od packets, we can simuate it as ide time). In tota, OPT has processed v k 1 tota vaue, and ALG has not processed more than 1 + v + v + v k which constitutes at most v k /v = v k of the unit cost of the heaviest packet. Now the minima competitive ratio for ALG out of the three cases is the ast one: v k 1 v k + 0 i k vi = v k 1 v k + vk 1 1 v 1 1 (v 1), and we take v = min{ B, B } to obtain the ower bound. Note that whie this bound is not too arge in practica cases, it is sti non-constant, that is, we now cannot hope for an onine agorithm with constant competitiveness uness we impose and use some additiona constraints on the probem setting. One fruitfu constraint turns out to be the constraint that the ony two aowed vaues are 1 and. I. UPPER BOUND FOR THE TO-ALUED CASE Given the pessimistic resuts of previous two sections, it remains ony to impose additiona constraints on at east one of the characteristic and try to distinguish important specia cases under which a good upper bound may exist. In this section, we consider an important specia case when there are ony two possibe packet vaues, 1 and, so there are two kinds of packets, (w 1) and (w ); the required processing can sti vary from 1 to. This case often occurs in practice; for instance, (w 1) may represent commodity packets whie (w ) corresponds to goden packets that have paid more to be processed. Simiar specia cases have been considered, e.g., in [3]. e wi show in Theorem 1 that in this specia case, the PQ v, w poicy has an attractive upper bound on the competitive ratio, which impies a constant upper bound on the competitive ratio of both PQ v, w and PQ v/w in case when <. This upper bound is fundamentay different from the ower bounds presented earier: instead of showing that an agorithm (or a set of those) sometimes performs bady, it shows that these particuar agorithms aways perform we. However, we begin with negative resuts; Theorem 11 provides matching tight ower bounds for the main resut that foows. A pot summarizing different ower bounds for the two-vaued case is shown on Figure. Theorem 11: Consider a buffer of size B with maxima required processing and possibe packet vaues 1 or. Then: (1) PQ w,v is at east -competitive; () if then PQ v/w is at east -competitive; (3) PQ v, w is at east ( + o(1)) -competitive. Proof: Lower bound Th 6, PQ f Th 7, Onine Th 11, PQ w,v Th 11, PQ v/w Th 11, PQ v, w Fig.. Lower bounds on the competitive ratios for the two-vaued case with fixed = 10. Higher vaue of a ower bound is better. (1) The construction from Theorem 1 uses ony packets of vaues 1 and. () Again, we present a hard sequence of arrivas. In the first burst, there arrive B (1 1) accepted by PQ v/w but not OPT. Then, on the same time sot there arrive B ( ) which PQ v/w has to miss since = 1 1 but which OPT accepts. Then, in B steps, PQ v/w wi have processed tota vaue B whie OPT wi have processed tota vaue B, which impies the bound. (3) In the first time sot, there arrive B ( ), which PQ v, w accepts, but OPT does not. Then, over the next time sots there arrives a (1 1) on every time sot. OPT transmits it, and PQ v, w drops it. On time sot +1, when PQ v, w has processed one ( ) packet, another ( ) packet arrives, which brings us back to the same state as on the first time sot. Over this sequence PQ v, w has processed packets with tota vaue, and OPT with tota vaue. Repeating this sequence C times, we get competitive ratio C+O(1) C+O(1) and et C. In the next theorem, we show one of the main resuts of this work, an upper bound on the competitive ratio of PQ v, w. Note that the ower bounds from Theorem 11 and previous sections, which were inear in, do not work for PQ v, w since in the two-vaued case, it aways processes packets with vaue first and with vaue 1 ast, so intuitivey we cannot ose more than the worst possibe packet with vaue, ( ), against the best possibe packet with vaue 1, (1 1) (Theorem 11 (3) shows that we cannot ose any ess). The foowing proof captures this intuition. Theorem 1: Consider a buffer of size B with maxima required processing and possibe packet vaues 1 or. Then PQ v, w is at most ( ) competitive. Proof: By the definition of the PQ v, w queue, any packet with vaue pushes out any packet with vaue 1. This is the crucia property that we need to prove this upper bound. For brevity, throughout this proof we denote PQ = PQ v, w. e define the foowing sets of packets:

9 9 (a) (b) a* p b* q X Y b ' b' +1 b* q a* p {a i '} {b i '} {a i '} {b i '} Fig. 3. Iustration for the subcases of Lemma 13: subcase p q is shown on (a), and subcase q < p is shown on (b). Arrows represent reation, and shaded areas denote sets of eements that sum to either A or B (note that emma s premise contains a A B inequaity); dotted ines denote specific position in a sequence. (1) IB ALG v = {p IB ALG : vaue(p) = v} contains packets with vaue v in IB ALG ; () Trans ALG v = {p transmitted by ALG : vaue(p) = v} is the set of packets with vaue v transmitted by ALG; Trans ALG = v TransALG = IB ALG v v ; (3) IBT ALG v is the set of packets with vaue v either aready transmitted by ALG or currenty residing in its buffer; IBT ALG = v IBTALG v. e aso define Φ ALG v () = i=1 w(p i), where p i is the ith Trans ALG v packet from IBT ALG v in PQ order. Here w(p) is the residua processing time of a packet at the current time moment; in particuar, we et w(p) = 0 for aready transmitted packets. In the proof, we wi sometimes force OPT to transmit certain packets immediatey, for free, thus improving its throughput. e denote the set of these packets at the current timesot as Free OPT ; they do not fa into Trans OPT but rather contribute to the objective separatey. The ony requirement is that for any p Free OPT we must have vaue(p) = 1, i.e., we ony give out packets of vaue 1 for free. e begin with a technica statement. Lemma 13: Let a 1, a,..., a m and b 1, b,..., b m be two sequences of numbers in nondecreasing order, and, moreover, suppose that {1,..., m} the prefix sums of ength satisfy the foowing inequaity: i=1 a i i=1 b i. Let aso a and b be any two numbers, such that a b. If a is inserted into a 1,..., a m, and b is inserted into b 1,..., b m then the prefix sums of resuting sequences satisfy the same inequaity. Formay, if the resut of a s insertion is a 1, a,..., a m+1 and the resut of b s insertion is b 1, b,..., b m+1, then we have, that {1,..., m + 1} i=1 a i i=1 b i. Proof: Denote right and eft hand sides of inequaities before (after) insertion as A and B (A and B ) respectivey. Let p and q be positions of a and b in the new sequences. Assume that < min{p, q}, then inequaities hod since none of the inserted vaues ie in the prefix of ength, consequenty, A = A and B = B. If max{p, q} then both inserted vaues ie in the prefix of ength, thus we have A = A 1 + a and B = B 1 + b, and it is easy to see that A B. The remaining case spits into two subcases (see Figure 3). p q. Denote X = i=p+1 a i. See next that A = A p 1 + a + X, B = B 1 + b, and aso A p 1 + X = A 1 B 1. Due to nondecreasing order: a b b, and we easiy get the required A B. q < p. Denote Y = i=p+1 b i. This gives us: A = A, B = B p 1 + b + Y, and we have A B = B p 1 + Y + b +1. Again, due to nondecreasing order: b +1 b, and caimed inequaity can be easiy derived. e now prove the crucia emma for this upper bound. Lemma 14: There exist an agorithm OPT that works no worse than the optima agorithm on any sequence of inputs and such a choice of Free OPT, that on every sequence of inputs at every time moment it hods that: (1) IBT PQ 1 IBT OPT 1, and for a, s.t. IBT OPT 1 it hods that Φ OPT 1 () Φ PQ 1 (); () IBT PQ IBTOPT, and for a, s.t. IBT OPT it () Φ PQ (); hods that Φ OPT Proof: e prove these estimates by induction on the number of events such as receiving, processing, or transmitting a packet. At the initia time moment a conditions hod triviay. Note that whenever conditions (1) and () hod, it aso necessariy hods that Trans OPT v Trans PQ v since Φ OPT v () = 0 for a Trans OPT v and consequenty Φ PQ v () = 0. e now remove from IBT PQ v the ( IBT PQ v IBT OPT v ) packets with the owest priority, denoting the resuting set by ĨBTPQ v and the corresponding set of packets in the buffer by ĨBPQ v. Then a of the above impies that IB OPT v ĨBPQ v. Upon acceptance we may mark a packet admitted to OPT buffer as causing overfow. The set of such packets is denoted as Over OPT, and it does not contribute to IBT OPT. The induction step wi guarantee that every packet in Over OPT is moved eventuay to Free OPT and the foowing invariant hods: Over OPT IB PQ ĨBPQ. Let us now consider a possibe events one by one and show that none of them vioates the conditions of the theorem. Arriva of a new packet p. There are two subcases. vaue(p) =. If PQ has accepted the packet and has pushed out a packet from IB PQ 1, we move the heaviest packet from IB OPT 1 (if it is nonempty) to Free OPT. Thus, inequaities for Φ 1 are not vioated since we have removed argest eements from both IB OPT 1 and IB PQ 1. Further, if OPT accepts p, then B > IB OPT ĨBPQ, so the sequence IBT PQ wi receive the ightest of packets (IB PQ \ ĨBPQ ) {p} (according to the push-out rues). Therefore, by Lemma 13 inequaities for Φ sti hod. The ony time IB PQ ĨBPQ increases is when IB PQ = B and OPT accepts, but IB OPT ĨBPQ and Over OPT + IB OPT < B together give Over OPT < IB PQ ĨBPQ. vaue(p) = 1. e consider two subcases separatey. (i) IB PQ + ĨBPQ 1 < B. In this case, we add to ĨBPQ 1 the ightest packet from (IB PQ 1 \ĨBPQ 1 ) {p} (according to push-out rues), and by Lemma 13 the inequaities are preserved. (ii) IB PQ + ĨBPQ 1 = B. Then, since

10 10 IB OPT ĨBPQ and IB OPT + Over OPT < B, we get that IB PQ ĨBPQ > Over OPT. Now, if OPT accepts p then p is added to the Over OPT. OPT processes a packet p. There are three subcases. vaue(p) =. This is a simpe case. If IB PQ then each nonzero term in Φ PQ () reduces exacty by one, whie each nonzero term in Φ OPT () reduces by at most one, so the inequaities are obviousy preserved. If otherwise IB PQ = then a Φ PQ = 0. vaue(p) = 1 and IB PQ =. Simiar to the previous. vaue(p) = 1 and IB PQ. In this case, p is sent to Free OPT. It remains to note that Φ OPT 1 () do not decrease since we have merey removed an eement from an ordered sequence. Transmitting a packet. Inequaities on Φ obviousy remain unchanged upon transmission; however, the vaue of ( IB PQ ĨBPQ ) can decrease by one. If Over OPT s invariant is vioated, we move an arbitrary packet from Over OPT to Free OPT. Lemma 15: The set Free OPT constructed in Lemma 14 satisfies the inequaity Free OPT ( + ) Trans PQ after agorithms finish processing the input sequence. Proof: It suffices to note that in the proof of Lemma 14 a packet may fa into Free OPT ony when PQ receives, processes, or transmits a packet with vaue. Now, after both OPT and PQ have processed the entire sequence of packets, the tota vaue of packets transmitted by PQ equas IBT PQ 1 + IBT PQ. The tota vaue of packets transmitted by OPT is IBT OPT 1 + IBT OPT + Free OPT. Thus, using the emma s inequaities, the competitive ratio α can be bounded as foows: α IBTOPT 1 + IBT OPT + Free OPT IBT PQ 1 + IBT PQ Free O P T 1 + IBT PQ 1 + IBT PQ 1 + Free O P T IBT PQ ( + ) IBTPQ 1 + IBT PQ ( Coroary ) 16: If <, PQ v, w and PQ v/w are at most + -competitive. Proof: Since ( ) pushes out (1 1) in the PQ v/w queue, any packet with vaue pushes out any packet with vaue 1, so for < PQ v/w is equivaent to PQ v, w. Figure 4 shows a contour pot of the PQ v, w competitive ratio upper bound 1+ + for the two-vaued case; naturay, for arge the bound is very good. II. THE β-push-out CASE In many networking systems, there arises an additiona motivation to avoid pushouts and prioritize packets that are aready in the buffer. For instance, newy admitted packets incur higher costs than packets that have aready resided in the buffer since they require more access bandwidth to packet memories: Fig. 4. Contour pot of the PQ v, w competitive ratio 1+ + as a function of and. a new packet incurs computationa costs for constructing and updating the corresponding data structures in the network processor, and immediate push-out of a ess-preferabe packet can ead to increased computationa overhead [14]. To represent these effects in the forma mode, Kesassy et a. [14] introduced the notion of copying cost in the performance of transmission agorithms for packets with heterogeneous processing requirements but uniform vaues: if an agorithm accepts A packets and transmits packets with tota vaue T, its transmitted vaue is max{0, T αa}, i.e., each admitted packet incurs a cost α subtracted from the throughput in the objective function. Thus, in extreme cases the transmitted vaue of a push-out poicy may even go down to zero; copying cost provides an additiona contro on the number of pushed out packets to avoid pathoogica cases. To impement such a contro mechanism, Kesassy et a. [14] introduced the greedy push-out work-conserving poicy P Q β that processes a packet with minima required work first and in the case of congestion such a poicy pushes out ony if a new arriva has at east β times ess work then the maxima residua work in PQ β. However, the work [14] ony deat with a singe packet characteristic, namey processing requirements. To generaize their ideas to our probem settings with two characteristics, we extend PQ f to PQ β f and then show severa ower bounds for β-push-out counterparts of our poicies. Unfortunatey, the proof of Theorem 1 cannot be directy appied to β- push-out poicies; it remains an interesting open probem to show nontrivia (ess than inear) upper bounds for β-push-out poicies even in the two-vaued case. Definition 7.1: Let f be a function of packets, f(w, v) R, with better packets corresponding to arger vaues of f. The PQ β f processing poicy for β > 1 is defined as PQ f with the foowing difference: PQ β f can push out a packet p and add a new packet p to the queue at time sot t if p is currenty the worst packet in the buffer and p is better than p at east by a factor of β: f(p) = min q IB PQ f f(q), and f(p ) > βf(p). 3 5

11 11 Theorem 17: Consider a buffer of size B with maxima required processing and maxima packet vaue. Then: (1) PQ β w,v is at east -competitive both in the case of arbitrary packet vaues and in the two-vaued case; () PQ β v/w is at east min{, }-competitive in the case of arbitrary packet vaues; (3) in the two-vaued case, if β then PQ β v/w is at east -competitive; ( ) (4) PQ β v, w is at east ( 1) o(1) -competitive in the case of arbitrary packet vaues and at east ( + o(1)) - competitive in the two-vaued case. Proof: (1) In the construction from Theorem 1, packets are never pushed out from PQ w,v buffer, so the resut sti hods for PQ β w,v. () Construction from Theorem 3 aso works because packets are never pushed out from PQ v/w buffer in tnis construction. (3) This resut can be seen as a reaxation of the second part of Theorem 11 since β > 1. The same construction works: PQ v/w fis its buffer with B (1 1) and then drops incoming B ( ). PQ β v/w aso drops them since β = β. (4) Again, the constructions from Theorem and Theorem 11 work since they do not force packets to be pushed out from PQ v,w buffer. III. SIMULATIONS In this section, we present the resuts of a comprehensive simuation study intended to vaidate our theoretica resuts. Naturay, it woud be desirabe to compare the proposed agorithms on rea ife network traces. Unfortunatey, avaiabe datasets such as CAIDA [10] are of itte use for packet characteristics in our mode since they do not provide data on required processing and intrinsic vaues of the packets. Nevertheess, we have used CAIDA traces [10] to mode the incoming stream of packets, breaking down the timestamps into equa timesots and counting the packets in each timesot; hence, the intensity of the incoming stream beow is measured in miiseconds, the size of a singe timesot. e have conducted six series of experiments, studying how performance depends on maxima required processing, buffer size B, maxima vaue, size of a CAIDA trace timesot t, and β (in the mode with copying cost). The actua optima onine agorithm in our mode woud be computationay prohibitive, so to estimate and compare the competitive ratios of our agorithms we have used an agorithm which is actuay better than optima: a singe priority queue with size B that breaks each packet (v w) into fractiona packets that each have required work 1 and vaue v w and then orders and processes them by this vaue. Since the priority queue has been proven optima in the mode with vaues and no required processing, it performs even better than optima. In our experiments, the vaues and processing ratios of packets were chosen uniformy from {1,..., } and {1,..., } respectivey. e ran a experiments for time sots with periodic fushouts (wait for a queues to finish their packets and then continue from an empty state), which in our experiments has proven to be sufficient for stabe resuts. e have aso performed simuations without fushouts; since the resuts are very cose to the ones with fushouts in a settings, we do not show them separatey. Note that a of our experiments venture into the vaues of parameters that yied high system oad with arge dropout rates for a agorithms; these are precisey the situations where we woud ike to compare performance since without heavy oad and frequent congestion a reasonabe agorithms perform identicay. e have made the code for our experimenta evauation pubicy avaiabe at GitHub [30]. Figure 5 shows simuation resuts presented in terms of the fraction of successfuy transmitted packets: each graph shows the better than optima reference agorithm in back aongside with the ratio of transmitted packets for other poicies. There are five sets of experiments corresponding to the rows of Fig. 5 that wi be described in subsections beow; we have tested the four agorithms used in this work: PQ v/w, w, PQ v/w,v, PQ w,v, and PQ v, w. Note that in a cases, PQ v/w, w and PQ v/w,v are virtuay indistiguishabe across a settings. Thus, beow we wi sometimes refer to them coectivey as PQ v/w. A. Maxima required processing In the first set of simuations (Fig. 5(1-3)), we study performance as a function of the maxima required processing. As grows, a agorithms deteriorate in absoute terms (packets become heavier), but it is cear that PQ w,v, which pays more attention to required processing, fares better whie PQ v, w oses bady as grows. This is expected since PQ v, w cares itte about and therefore is ikey to get stuck with very heavy packets. e see that PQ v/w is uniformy the best poicy, performing very cose to OPT and deteriorating ony sighty. B. Buffer size B In the second set of simuations (Fig. 5(4-6)), we study performance as a function of the buffer size B. In this setting as we, PQ v/w, w and PQ v/w,v remain indistinguishabe, and since these experiments were done in the reativey ow ranges of the / ratio, PQ v, w is aso very cose to PQ v/w. PQ v, w, on the other hand, is abe to store, in a arger buffer, more high-vaue packets and do so for onger, so as B increases and congestion decreases, PQ v, w becomes coser to the other three. Note, in this setting, a agorithms become significanty worse off compared to the fractiona OPT through no faut of their own: the unfairness of fractiona OPT becomes much more pronounced with arge B (it has more and more extra space to store packets). C. Maxima vaue In the third set of experiments (Fig. 5(7-9)), we ook at performance as a function of the maxima vaue. It turns out that whie the reative performance to fractiona OPT drops, the performance eve of the three eading agorithms does not significanty depend on in reaistic cases, and ony PQ v, w drops significanty in reative quaity. This, again, can be expained by the fact that PQ v, w suffers from a wider variety of packets, getting stuck with vauabe yet heavy ones. The reative order of agorithms remains unchanged.

12 1 D. Incoming stream intensity The fourth set of experiments (Fig. 5(10-1)) shows how performance depends on the intensity of the packet source, expressed in terms of the timesot size t (naturay, more packets on average arrive during a onger t). This setting ets us expore the most congested settings: a agorithms join together at the high end of intensity simpy because now there are, on average, enough (1 ) packets arriving to keep a agorithms busy ony with the obviousy best packets. The reative standings of a agorithms remain the same throughout this increasing congestion. E. β for β-push-out poicies The fifth set of experiments (Fig. 5(13-15)) studies a different situation; here, we have introduced nonzero copying cost α (on a three graphs, α = 0.3) and have studied how performance depends on β for β-push-out counterparts of our poicies (as introduced in Section II); since the number of admitted packets is not we defined for our fractiona OPT, OPT did not participate in these experiments; we have taken the resuts of PQ v/w, w for β = 1 as the starting point, dividing a the rest by this vaue. e see that in a cases, β = 1 appears to be the perfect or amost perfect choice in practice: sometimes β = 1. or β = 1.3 yied better resuts, but ony sighty. F. for the two-vaued case The ast, sixth set of experiments (Fig. 5(16-18)) deas with the two-vaued case, when the intrinsic vaue of a packet can ony take vaues in {1, } whie required work can sti be an arbitrary integer from 1 to. e repeated the experiments from Section III-A with this additiona restriction, and the resuts cosey match our theoretica resuts from Section I: contrary to the genera case, now PQ v, w is not the obviousy worst agorithm but performs on par with PQ v/w poicies, whie PQ w,v diverges from them for arger in exacty the same way as in the genera case (compare to Section III-A, Fig. 5(1-3)). Since PQ v, w may be easier to impement than PQ v/w (required work does not have to be considered or even known), for the two-vaued case we recommend to use PQ v, w. Again, as a side effect we see that the fractiona OPT performs better (reative to other agorithms) when more buffer space is provided. To summarize, in this section we have shown a comprehensive simuations study on synthetic traces. The main resut is that the PQ v/w poicy that we have introduced in this work is uniformy the best poicy across a tested settings, and there is itte difference between tie-breaking variations of it, whie in the two-vaued case experimenta resuts supported the theoretica concusion that PQ v, w is a good poicy. IX. CONCLUSION In this work, we have begun the study of buffer management for processing packets with two different characteristics: processing requirement and vaue. In these settings we have considered a singe queue buffering architecture and have Processing poicy Genera case Two-vaued case Adversaria genera ower bounds min{ Any onine agorithm, B } O ( ) 1 Any priority queue ( ) min{, / } Any FIFO onine agorithm B B 1 B +B 1 Lower and upper bounds for specific agorithms Lower bound Lower Upper PQ w,v, PQ β w,v PQ v, w, PQ β ( 1) + v, w o(1) + o(1) 1 + PQ v/w, PQ β v/w, β PQ v/w, < + o(1) + TABLE I RESULTS SUMMARY: LOER AND UPPER BOUNDS. mosty studied agorithms based on priority queues; in the setting with two characteristics, there may be different reasonabe priority queues that have different poicies. e have investigated various packet processing orders and found that they have inear ower bounds on the competitive ratio, which makes them unattractive in the genera case. However, we have provided positive resuts in the specia case of two different vaues, 1 and and heterogeneous processing requirements. The resuts of our work are summarized in Tabe I; note that a agorithms in the tabe empoy push-out (abeit with different heuristics for it). In the main resut of this work, we have shown a (1 + ( + )/ ) upper bound for the buffer management poicy PQ v, w that orders packets first by vaue and then by required processing. For <, this aso becomes a constant upper bound on the competitive ratio of PQ v/w which orders packets by unit processing (ratio of vaue to processing). This resut has been somewhat counterintuitive since the intuition woud be that the PQ v/w poicy that optimizes for vaue per timesot woud be best, but in the genera two-vaued case it has a non-competitive ower bound. In addition, we have shown a number of genera ower bounds, for the cases of any deterministic onine agorithm with FIFO processing and transmission order, for any priority queue, and even for any deterministic onine agorithm at a; whie these ower bounds are reativey weak, they are nonconstant and show that it is impossibe to achieve constant upper bounds in these cases without additiona assumptions on the reations between parameters such as B,, and. For the two-vaued case, we have shown tighty matching ower and upper bounds on the competitive ratio (they differ by 1+o(1)). It sti remains an interesting open probem to prove upper bounds for the genera case of arbitrary vaues; another interesting probem woud be to prove upper bounds for β- push-out poicies. However, the reay crucia question here is whether there exists a processing poicy with better than inear competitive ratio for the genera case of two characteristics: our genera ower bounds are not constant but they are far from inear too. e suggest this probem for further study. Acknowedgements: e thank both IEEE INFOCOM 015 and IEEE/ACM Transactions on Networking reviewers for their insightfu comments. This work was supported by the Russian Science Foundation grant Networking and distributed systems and agorithms and reated fundamenta probems.

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Anaysis of queueing poicies in QoS switches. J. Agorithms, 53(): , 004. Pave Chuprikov Pave Chuprikov is a research assistant at the IMDEA Networks Institute, Spain and a Ph.D. student at the Stekov Institute of Mathematics at St. Petersburg, Russia. Previousy, he worked as a software deveoper at JetBrains Research. Pave received his M.Sc from St. Petersburg Academic University of Russian Academy of Sciences. His research interests incude software defined networking, onine agorithm design, and dependent types. Aex Davydow Aex Davydow is a researcher at the Stekov Institute of Mathematics at St. Petersburg. He obtained his M.Sc. from the St. Petersburg Academic University and graduated from its Ph.D. studies. His research interests incudes networking agorithms and systems, tropica agebraic geometry and its appication to scheduing agorithms. Sergey Nikoenko Sergey Nikoenko is a researcher at the Stekov Institute of Mathematics at St. Petersburg (PDMI RAS). He received his M.Sc. summa cum aude from St. Petersburg State University (005); Ph.D., from PDMI RAS (009). His research interests incude networking agorithms and systems, machine earning and probabiistic inference, bioinformatics, and theoretica CS, with projects funded by major companies and research funds such as RSF, CRDF, INTAS, Mai.Ru, RFBR, RAS, and others. Kiri Kogan Kiri Kogan is a Research Assistant Professor at IMDEA Networks Institute. He received his PhD from Ben-Gurion University (Israe) at 01. He was a Technica Leader at Cisco Systems, where he worked in He was a Postdoctora Feow at University of ateroo and Purdue University during His current research interests are in design, anaysis, and impementation of networked systems, broady defined.